Surfaces in Non-flat 3-Space Forms Satisfying square vec {textbf{H}}=lambda vec {textbf{H}}

Abstract

In this paper, we locally classify the surfaces immersed into the non-flat (Riemannian or Lorentzian) 3-space forms satisfying the condition Box vec {textbf{H}}=lambda vec {textbf{H}} for a real number lambda , where vec {textbf{H}} is the mean curvature vector field and Box denotes the Cheng–Yau operator of the surface. We obtain the classification result by proving, at a first step, that the mean curvature function must be constant and, in a second step, we complete the classification.